Is the sequence $$a(n)=\frac{\sin{(3^{n})}}{\sin{(5^{n})}}$$ convergent or divergent? As $n$ tends to $\infty$, I am not getting which limit formula to apply.
Any hint?
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there is no Limit (infinity) – Dr. Sonnhard Graubner Nov 18 '17 at 20:46
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the formula above – Dr. Sonnhard Graubner Nov 18 '17 at 20:48
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both function do not have limits – Dr. Sonnhard Graubner Nov 18 '17 at 20:52
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1But both functions are bounded by 1 so I am not satisfied yet . – ogirkar Nov 18 '17 at 20:59
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2@Dr.SonnhardGraubner You could say the same about $\lim_{n\to\infty}\frac{\sin(3^n)}{\sin(3^n)}$, but in that case the limit is clearly $1$. – Arthur Nov 18 '17 at 21:07
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I am puzzled by this question, it looks more like a number-theoretic question about irrationality of numbers of type $\frac{3^n}{\pi}$ (or $\frac{5^n}{\pi}$) than a calculus question. I cannot answer it personally. Out of curiosity, what is the context where you came up with this question? – Nov 18 '17 at 21:10
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Reminds me of this thread although this question might be easier to resolve(?). – Jeppe Stig Nielsen Nov 18 '17 at 21:19
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@Jeppe Stig Nielsen..thanks for the link.. – ogirkar Nov 18 '17 at 21:40
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@user8734617 I cannot solve it either, I think. While it is known that the numbers you mention are irrational, that is not enough. – Jeppe Stig Nielsen Nov 18 '17 at 21:41
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What book (author(s), title)? – Jeppe Stig Nielsen Nov 18 '17 at 21:45
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It's not reference book..just local maths book by one publication.. – ogirkar Nov 18 '17 at 21:47
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@JeppeStigNielsen I know they are irrational, but what I really meant is that any proof may draw from the theory of Diophantine approximations (https://en.wikipedia.org/wiki/Diophantine_approximation) which is used to measure how 'close' to rational are irrational numbers. Thus I wonder if this post merits cross-posting to MathOverflow. – Nov 18 '17 at 22:36